等时曲线是摆线的证明  

The Prove of Isochronous Curve Being Cycloid

作　　者：邢家省[1,2] 杨义川[1,2] 王拥军[1,2] XING Jiasheng;YANG Yichuan;WANG Yongjun(School of Mathematics, Beihang University, Beijing 100191,China;LMIB of the Ministry of Education, Beihang University, Beijing 100191 , China)

机构地区：[1]北京航空航天大学数学与系统科学学院,北京100191 [2]数学信息与行为教育部重点实验室,北京100191

出　　处：《四川理工学院学报（自然科学版）》2019年第2期90-94,共5页Journal of Sichuan University of Science & Engineering(Natural Science Edition)

基　　金：国家自然科学基金(11771004);北京航空航天大学校级重大教改项目:北航2016年数学重点教改项目

摘　　要：为了考虑等时曲线的求解问题,建立质点沿光滑曲线从一定高度下滑所需时间的公式,将该问题转化为一个积分方程的求解问题。对无限区间上的积分方程,利用拉普拉斯变换方法给出了求解方法,得到了积分方程解的解析表达式,然后将其变化为一个常微分方程的求解问题。对有限区间上的积分方程,利用含参变量积分的求导和积分交换次序方法,得到积分方程解的解析表达式。然后将等时曲线问题,转化为一个常微分方程的求解问题,通过求解得到等时曲线解的解析表达式,即摆线的方程形式,从而给出了具有等时性的曲线一定是摆线的证明过程,对等时曲线的问题给予了完整的解决。In order to solve the problem of isochronous curve,a formula for the time required for the particle to slide down from a certain height along a smooth curve is established,and the problem is transformed into an integral equation. For the integral equation in infinite interval,an analytical expression of the solution of the integral equation is obtained by using the Laplace transform method. And then changed into an ordinary differential equation. For the intergral equation in finite interval,the analytic expression of the solution of the integral equation is obtained by using the method of derivative of integral with parameter variable and the method of integral exchange order,and then the isochronous curve problem is transformed into an ordinary differential equation. Analytical expression of isochronous curve is obtained by solution and it happens to be the equation form of the cycloid. It is shown that the isochronous curve must be the proof process of cycloid,and the problem of isochronous curve is completely solved.

关 键 词：等时曲线 摆线 积分方程 拉普拉斯变换 常微分方程 

分 类 号：O175.29[理学—数学]